Non-negative Matrix Factorization
to identify Latent Factors
Underlying Psychopathology

Laura Sità

Problem

Classification of psychopathology

Traditional taxonomies: categorical diagnostic systems

(e.g., DSM, ICD)

Recent models: dimensional and transdiagnostic approaches

(e.g. Hierarchical Taxonomy of Psychopathology model by Kotov et al., 2017)

Research Question

How can we uncover latent factors across disorders?

  • Factor Analysis
  • Proposed approach (inspired by Landy et al., 2025): Non-negative Matrix Factorization (NMF)

Proposed approach

NMF

NMF factorizes the full observed data matrix

\(M_{kg} \in \mathbb{R}_{\ge 0}^{K \times G}\),
where each row \(k\) represents an observed feature (e.g., test item)
and each column \(g\) represents an individual.

It decomposes \(M\) into two lower-rank nonnegative matrices:

\(P \in \mathbb{R}_{\ge 0}^{K \times N}\) → loadings of observed variables on \(N\) latent factors

\(E \in \mathbb{R}_{\ge 0}^{N \times G}\) → expression (or weights) of latent factors across individuals

\[ M_{kg} = \sum_{n=1}^{N} P_{kn} E_{ng} \]

First step

  • Use the NMF on a dataset of ordinal data (Likert scale)

  • Estimate latent factors

  • Compare them with those obtained through factor analysis

codice

mostra il codice sul dataset di tommaso

matrici item x fattori tramite EFA vs NMF

matrici item x fattori tramite CFA vs NMF

Second step

how to make the NMF recognize that items within the same questionnaire have a structure

Estensions

If the results are encouraging …

  • Third step: use NMF to find latent factors shared within the same spectrum of symptoms

  • Fourth step: use causal NMF to find latent factors across different treatment on the same spectrum of symptoms

Materials

All materials are available on GitHub at laurasitaunipd/nmf

Bibliography

Kotov, R., Krueger, R. F., Watson, D., Achenbach, T. M., Althoff, R. R., Bagby, R. M., … & Zimmerman, M. (2017). The Hierarchical Taxonomy of Psychopathology (HiTOP): A dimensional alternative to traditional nosologies. Journal of abnormal psychology, 126(4), 454.

Landy, J. M., Basava, N., & Parmigiani, G. (2025). bayesNMF: Fast Bayesian Poisson NMF with Automatically Learned Rank Applied to Mutational Signatures. arXiv preprint arXiv:2502.18674.

Landy, J. M., Zorzetto, D., De Vito, R., & Parmigiani, G. (2025). Causal Inference for Latent Outcomes Learned with Factor Models. arXiv preprint arXiv:2506.20549.

Supplemental Materials